H-infinity control with integrator compensation for anode pressure control in a fuel cell stack

ABSTRACT

Pressure control in a fuel cell is achieved by using an H-infinity controller coupled in a feedback loop between a reactant feed gas valve and a pressure sensor on gas flows to the membrane electrode assembly of the fuel cell. To maintain pressure balance across the membrane, the pressure of the oxidant reactant is used to regulate fuel reactant flow. An integrator windup compensator manages integral windup in the H-infinity control scheme. Control weight, sensor noise weight, and performance weight matrices are incorporated into the H-infinity control model. Respective to PID control, the H-infinity model provides superior performance in the presence of high frequency feedback noise enabling use of low cost control components in the fuel cell and a minimum of EMI shielding.

FIELD OF THE INVENTION

[0001] The present invention relates to fuel cell power systems andmethods for controlling pressure in a reactant feed gas stream to a fuelcell stack of the fuel cell power system.

BACKGROUND OF THE INVENTION

[0002] Fuel cell power systems convert a fuel and an oxidant toelectricity. One fuel cell power system type of keen interest employsuse of a proton exchange membrane (hereinafter “PEM”) to catalyticallyfacilitate reaction of fuels (such as hydrogen) and oxidants (such asair/oxygen) into electricity. The PEM is a solid polymer electrolytethat facilitates transfer of protons from the anode to the cathode ineach individual fuel cell of the stack of fuel cells normally deployedin a fuel cell power system.

[0003] In a typical fuel cell assembly (stack) within a fuel cell powersystem, individual fuel cells have flow fields with inlets to fluidmanifolds; these collectively provide channels for the various reactantgases and cooling fluids in the stack to flow into each cell. Gasdiffusion assemblies then provide a final fluid distribution to furtherdisperse reactant fluids from the flow field space to the reactive anodeand cathode; these diffusion sections are frequently advantageouslyembedded as a part of the design of collector electrodes pressingagainst the reactive anode and cathode.

[0004] Effective operation of a PEM requires maintenance of a smallpressure drop between the cathode (air) and anode (hydrogen) gasesacross the PEM; in this regard, accurate pressure control is vital tofuel cell stack performance and durability.

[0005] Control of fuel cell power systems must also resolve highfrequency noise derived from EMI (electromagnetic interference); sourcesof EMI are both internal from the components of the fuel cell as well asexternal, especially when the fuel cell powers a vehicle which movesfrom place to place and thereby experiences different EMI environments.

[0006] There is an ongoing desire to minimize cost in fuel cell systems.Low cost components (such as pressure and feed control valves), however,frequently demonstrate susceptibility to EMI and also provide marginalacceptability in maintaining acceptably balanced pressures in fuel cellstacks when used with traditional PID (proportional-integral-derivative)control schemes. Components (such as pressure and feed control valves)which demonstrate good resistance to EMI and also provide acceptabilityin maintaining balanced pressures in fuel cell stacks when used withtraditional PID (proportional-integral-derivative) control schemes arenot favored for deployment because of higher cost.

[0007] What is needed is an approach to fuel cell pressure control whichprovides acceptable precision in balancing pressures across a PEM at lowcost. The present invention is directed to fulfilling this need.

SUMMARY OF THE INVENTION

[0008] The present invention provides pressure control in a fuel cellhaving at least one membrane electrode assembly in reactive interface(a) to a plurality of oxidant reactant flow channels carrying an oxidantreactant and (b) to a plurality of fuel reactant flow channels carryinga fuel reactant, using: a valve disposed to control at least onereactant flow to the membrane electrode assembly; a pressure sensordisposed to measure pressure within the fuel cell; and an H-infinitycontroller coupled in a feedback loop between the valve and the pressuresensor.

[0009] As a method, the invention operates a fuel cell having at leastone membrane electrode assembly in reactive interface (a) to a pluralityof oxidant reactant flow channels carrying an oxidant reactant and (b)to a plurality of fuel reactant flow channels carrying a fuel reactantby measuring pressure within the fuel cell; deriving a setpoint for atleast one reactant flow from an H-infinity control model in response topressure data from the measuring step; and regulating each reactant flowfor which the deriving step derives a setpoint.

[0010] The invention further provides that the pressure of the oxidantreactant is used to regulate fuel reactant flow.

[0011] The invention also provides for use of an integrator windupcompensator in data communication with the H-infinity controller andalso for use of a real-time computer to execute the H-infinitycontroller and/or the windup compensator.

[0012] The invention further provides for incorporation of (a) a controlweight matrix, (b) a sensor noise weight matrix, and/or (c) aperformance weight matrix in the H-infinity control model.

[0013] When compared to a standard PID controller, the inventionprovides enhanced performance in the presence of high frequency feedbacknoise to provide an improved operation of the control valve, lesspart-to-part actuator variation, and reduced system retuning. Theinvention further enables use of low cost control components in the fuelcell and minimizes the amount of EMI shielding needed for effectivepower generation.

[0014] Further areas of applicability of the present invention willbecome apparent from the detailed description provided hereinafter. Itshould be understood that the detailed description and specificexamples, while indicating the preferred embodiment of the invention,are intended for purposes of illustration only and are not intended tolimit the scope of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

[0015] The present invention will become more fully understood from thedetailed description and the accompanying drawings, wherein:

[0016]FIG. 1A presents a first embodiment of a fuel cell power systemincorporating the present invention;

[0017]FIG. 1B presents a second embodiment of a fuel cell power systemincorporating the present invention;

[0018]FIG. 2 shows membrane electrode assembly detail in a fuel cellstack portion;

[0019]FIG. 3 shows the anode pressure due to a step response of aproportional flow control valve;

[0020]FIG. 4 shows variance between a model of the step responseaccording to FIG. 3 and measured data;

[0021]FIG. 5 shows a feedback control system model in a form consistentwith robust analysis;

[0022]FIG. 6 depicts a patterned schema showing management ofuncertainties in a control loop model;

[0023]FIG. 7 provides a frequency response profile plot of a transferfunction related to a control weight matrix;

[0024]FIG. 8 provides a frequency response profile plot of a transferfunction related to a sensor noise weight matrix;

[0025]FIG. 9 provides a frequency response profile plot of a transferfunction related to a performance weight matrix;

[0026]FIG. 10. shows an exemplary feedback loop in overall linearfractional transformation form;

[0027]FIG. 11 provides detail in a state-space matrix of the linearfractional transformation form of the exemplary control problem reviewedin this specification; and

[0028]FIG. 12 shows detail in an H-infinity controller with anintegrator windup compensation block.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0029] The following description of the preferred embodiment is merelyexemplary in nature and is in no way intended to limit the invention,its application, or uses.

[0030] In utilizing a low cost proportional flow control valve tomaintain a balanced pressure across a PEM, the anode pressure setpointand/or fuel flow to the anode is determined from the cathode pressureagainst a user selected pressure set-point. In the past, standard PIDcontrol has been used for such control along with supporting feedforwardloops and software filters. However, H-infinity (H^(∞)) control of thisapplication is shown herein to be well suited as a robust controlapproach. Details in the methodology of H-infinity control areestablished in the art and are appreciated from a study of “Essentialsof Robust Control” by Kermin Zhou and John C. Doyle (Prentice Hall,1998). The discussion of the preferred embodiments also references twoadditional concepts as further detailed in the following two paragraphs.

[0031] The discussion of the preferred embodiments references theconcept of “steady-state” operation. As used herein, “steady-state”operation or “steady state” is considered a situation where (1) aprocess is dynamically regular and uniform in its operation over a timeinterval, (2) momentum, mass, and energy entities flowing into theprocess are essentially equal to the momentum, mass, and energy entitiesflowing out of the process (excepting anticipated accumulations), and(3) accumulations of momentum, mass, and energy within the process areessentially not occurring unless they are explicitly expected andfactored into the relevant dynamic model. Mathematical solutions of thebalances with respect to the status of steady state operation need toalso accommodate expected chemical reactions. Steady state operation ofa system is an issue of importance to the present invention sincecertain of the modeling equations are based upon the presumption thatreal-time input data used in a specific instance of a control decisionhave a collective associated steady state relationship. A system in“steady state” therefore, has attributes of dynamic balance, stability,steadiness, and equilibrium.

[0032] The concept of real-time computer process control is also auseful term in understanding the preferred embodiment. As used herein,real-time computer processing is broadly considered as a method ofcomputer processing in which an event causes a given reaction within anactual time limit and wherein computer actions are specificallycontrolled within the context of and by external conditions and actualtimes. As an associated clarification in the realm of process control,real-time computer controlled processing relates to the performance ofassociated process control logical, decision, and quantitativeoperations intrinsic to a process control decision program functioningas part of a controlled apparatus implementing a process (such as thefuel cell benefiting from the present invention) wherein the processcontrol decision program is periodically executed with relatively highfrequency—e.g., having a period of between 20 ms and 2 sec for highlytactical control, or on the order of 10 to 100,000 times the period ofthe associated tactical control decision frequency for “on line”real-time advanced control routines, simulators, and optimizers, withoutlimitation. The larger period for advanced control routines, simulators,and optimizers is frequently necessary to accommodate the substantialcomputer calculations which must be performed within one decision cycleof the advanced control routine, simulator, or optimizer. With furtherregard to the time period during which the process control decisionprogram is periodically executed, some operations are optionallyperformed on a multiple of the process control decision programexecution period needed for computation time; this less frequentoperation period is usually adopted for purposes related to tuning,sensitivity, and efficient resource utilization.

[0033] The invention is further understood with reference to a genericfuel cell power system. Therefore, before further describing theinvention in detail, a general overview of the types of power systemswithin which the present invention operate is provided. Reference ismade to hydrogen-containing as having relatively high hydrogen content.The invention is hereafter described in the context of a fuel cellfueled by an H₂-containing reformate regardless of the method by whichsuch reformate is made. It is to be understood that the principlesembodied herein are applicable to fuel cells fueled by H₂ obtained fromany source, including reformable hydrocarbon and hydrogen-containingfuels such as methanol, ethanol, gasoline, alkaline, or other aliphaticor aromatic hydrocarbons.

[0034] A first preferred system 100 a illustrated in FIG. 1A includeshydrogen source 102 such as a hydrogen storage tank and oxidant source104 such as ambient air provided via a pump or compressor (not shown).Hydrogen source 102 directs H₂-containing feed stream 120 to the anodeside of fuel cell 122. Oxidant source 104 directs O₂-containing feedstream 124 to the cathode side of fuel cell 122. Anode exhaust (oreffluent) 126 is discharged from the anode side of fuel cell stacksystem 122. Cathode exhaust (or effluent) 128 is discharged from thecathode side of fuel cell stack system 122. Pressure of cathode exhaust128 from the cathode side of fuel cell stack system 122 is measured bypressure sensor 160.

[0035] A second preferred system 100 b illustrated in FIG. 1B includes afuel processor 112 for catalytically reacting a reformable hydrocarbonfuel stream 114, and water in the form of steam from a water stream 116.In some fuel processors, air is also used in a combination partialoxidation/steam reforming reaction. In this case, fuel processor 112also receives an air stream 118. The fuel processor 112 contains one ormore reactors wherein the reformable hydrocarbon fuel in stream 114undergoes dissociation in the presence of steam in stream 116 and air instream 118 to produce the hydrogen-containing reformate exhausted fromfuel processor 112 in reformate stream 120. Fuel processor 112 typicallyalso includes one or more downstream reactors, such as water-gas shift(WGS) and/or preferential oxidizer (PrOx) reactors that are used toreduce the level of carbon monoxide in reformate stream 120 toacceptable levels, for example, below 20 ppm.

[0036] Anode exhaust (or effluent) 126 is discharged from the anode sideof fuel cell stack system 122 and may contain some unreacted hydrogen.Cathode exhaust (or effluent) 128 is discharged from the cathode side offuel cell stack system 122 and may contain some unreacted oxygen.Pressure of cathode exhaust 128 from the cathode side of fuel cell stacksystem 122 is measured by pressure sensor 160. These unreacted gasesrepresent additional energy recovered in combustor 130, in the form ofthermal energy, for various heat requirements within power system 100.Specifically, a hydrocarbon fuel 132 and/or anode effluent 126 arecombusted, catalytically or thermally, in combustor 130 with oxygenprovided to combustor 130 either from air in stream 134 or from cathodeeffluent stream 128, depending on power system 100 operating conditions.Combustor 130 discharges exhaust stream 154 to the environment, and theheat generated thereby is directed to fuel processor 112 as needed.

[0037] In both embodiments illustrated in FIGS. 1A and 1B, H₂-containingreformate 120 is fed through control valve 162 into the anode chamber offuel cell stack system 122. Control valve 162 may be either an analogcontrol valve or a solenoid spring return valve similar to a fuelinjector valve with a 100 Hz duty cycle frequency. Concurrent with thefeeding of H₂-containing reformate 120 through control valve 162 intothe anode chamber of fuel cell stack system 122, oxygen in the form ofair in stream 124 is fed into the cathode chamber of fuel cell stacksystem 122. The hydrogen from reformate stream 120 and the oxygen fromoxidant stream 124 react in fuel cell stack system 122 to produceelectricity.

[0038] Real-time computer 164 effects control of valve 162 in responseto a signal from at least pressure sensor 160. That is to say the anodefeed gas is controlled through use of real-time computer 164 withrespect to the pressure of the cathode oxidant gas in fuel cell 122,although other parameters may also be utilized in the control of theanode feed gas. Controller logic 166 is provided for execution inreal-time by computer 164. As presently preferred, controller logic 166is also denoted as “software” and/or a “program” and/or an “executableprogram” within real-time computer 164 as a data schema holding dataand/or formulae information and/or program execution instructions.Controller logic 166 is, in a preferred embodiment, machine coderesident in the physical memory storage (i.e., without limitation,random access memory having “RAM” as an indicator, read only memoryhaving “ROM” as an indicator, or a computer disk) of computer 164.Controller logic 166 is preferably derived from a source languageprogram compiled to generate the machine code. The physical memorystorage is in electronic data communication with a central processingunit (CPU) of computer 164 which reads data from the physical memory,computationally modifies read data into resultant data, and writes theresultant data to the physical memory. Computer 164 also read signalsfrom sensor 160 and effects signals to valve 162 according to theprovisions of controller logic 166.

[0039] The fuel cell power systems described abve include a hydrogenstorage-based system or a fuel reforming system. Thus, a skilledpractitioner will recognize that the present invention has applicationto a variety of system which obtain fuel from diverse sources. In thisregard, the manner in which the fuel is generated does not impact thepresent invention or its application into a fuel cell power system.

[0040] Turning now to FIG. 2, a partial PEM fuel cell stack 200 of fuelcell stack system 122 is schematically depicted as having a pair ofmembrane electrode assemblies (MEAs) 208 and 210 separated from eachother by a non-porous, electrically-conductive bipolar plate 212. Eachof MEAs 208, 210 have a cathode face 208 c, 210 c and an anode face 208a, 210 a. MEAs 208, 210 and bipolar plate 212 are stacked togetherbetween non-porous, electrically-conductive, liquid-cooled end plates214 and 216. Plates 212, 214, 216 each include respective flow fields218, 220, 222 established from a plurality of flow channels formed inthe faces of the plates for distributing fuel and oxidant gases (i.e.,H₂ & O₂) to the reactive faces of MEAs 208, 210. Nonconductive gasketsor seals 226, 228, 230, 232 provide sealing and electrical insulationbetween the several plates of fuel cell stack 200.

[0041] Porous, gas permeable, electrically conductive sheets 234, 236,238, 240 press up against the electrode faces of MEAs 208, 210 and serveas primary current collectors for the respective electrodes. Primarycurrent collectors 234, 236, 238, 240 also provide mechanical supportsfor MEAs 208, 210, especially at locations where the MEAs are otherwiseunsupported in the flow field. Bipolar plate 214 presses up againstprimary current collector 234 on cathode face 208 c of MEA 208, bipolarplate 216 presses up against primary current collector 240 on anode face210 a of MEA 210, and bipolar plate 212 presses up against primarycurrent collector 236 on anode face 208 a of MEA 208 and against primarycurrent collector 238 on cathode face 210 c of MEA 210.

[0042] An oxidant gas such as air/oxygen is supplied to the cathode sideof fuel cell stack 200 from air source 118 and line 124 via appropriatesupply plumbing 242. In a preferred embodiment, air is supplied to thecathode side from the ambient. A fuel such as hydrogen is supplied tothe anode side of fuel cell 200 from fuel source 270 via appropriatesupply plumbing 244. In a preferred embodiment, the fuel source issupplied from a reformer via line 120 after catalytically dissociatinghydrogen from hydrocarbon fuel 114.

[0043] Exhaust plumbing (not shown) for both the H₂ and O₂/air sides ofMEAs 208, 210 is also provided for removing anode effluent from theanode flow field and the cathode effluent from the cathode flow field.Coolant plumbing 250, 252 is provided for supplying and exhaustingliquid coolant to bipolar plates 212, 214, 216, as needed.

[0044] It is to be noted that fuel cell stack 200 shows two fuel cellswith plate 212 being shared between the two fuel cells and plates 214,216 being shared between one of the shown fuel cells and, in each case,another fuel cell not depicted in FIG. 2.

[0045] Turning now to further detail in controller logic 166 ofreal-time computer 164 and with reference to FIG. 1, fuel cell powersystem 100 uses proportional flow control valve 162 to control reactantfeed gas flow, and pressure sensor 160 is used as a feedback sensormeasuring cathode gas pressure of fuel cell 122. A robust H-infinitycontroller is effected in controller logic 166 in a feedback loopbetween valve 162 and pressure sensor 160.

[0046] In overview, the first step of the present invention is to obtaina fundamental dynamic response model from the control loop defined frompressure sensor 160, real-time computer 164, and control valve 162. Aswill be described further herein, a first-order model empiricallycharacterizes the dynamic relationship of the feedback loop betweenvalve 162 and pressure sensor 160 in fuel cell power system 100 for theexemplary control loop and derived H-infinity robust controller. In thisregard, a standard discrete time system identification technique,AutoRegressive eXogeneous or ARX, determines the first-orderrelationship from numerous open-loop responses of pressure 160 to stepchanges in control valve 162 and derives a set of models encompassingthe essential full range of anticipated behavior for fuel cell powersystem 100. Response model uncertainties may be further determined basedon the standard deviation of the models found during each responsemeasurement.

[0047] The second step relates to development of the H-infinity dataschema for the H-infinity controller. The dynamic response model withuncertainties (from the first step of the overview) is combined withvarious weighting data. For example, dynamic response noise “weighting”data (e.g., a noise weight matrix) derived from measurements of knownhigh frequency EMI (electromagnetic interference) feedback noise iscombined with the combined data derived from the design and measurementsof the closed control loop and configured into a Linear FractionalTransformation (LFT) framework. Dynamic controller response “weighting”data (e.g. a control weight matrix) and dynamic response plantperformance “weighting” data (e.g. a performance weight matrix) may befurther incorporated into the LFT framework. An H-infinity data schemafor an H-infinity real-time controller is then calculated from the LFTframework. Thus, the result of the first step is a dynamic responsemodel which includes data set(s) derived from the design andmeasurements of the closed control loop.

[0048] Insofar as a controller derived solely from the H-infinity dataschema will have an integral windup effect, a integrator compensationgain may also be effected in the H-infinity controller of controllerlogic 166 in real-time computer 164 to provide an integrator windupcompensation block in controller logic 166, and thereby providereal-time modification of the output from real-time calculations of theH-infinity data schema to compensate for integrator windup in thecontrol loop and maintain control valve 162 in an immediately responsivestatus.

[0049] Lastly, the robust control data schema and integrator windupcompensator are operably incorporated in controller logic 166 ofreal-time computer 164 which is employed to control fuel cell powersystem 100.

[0050] Turning now to greater detail in implementing the steps describedin the above overview, FIGS. 3 and 4 show step response 300 andempirical data and model comparison 306, respectively the curvesillustrated in graphs 300, 400 may be derived (either directly of agiven fuel cell or via a proximate pressure vessel physical modelthereof) from the exemplary feedback loop between valve 162 and pressuresensor 160 in fuel cell power system 100 through use of a conventionalsystem identification technique for discrete data such as AutoRegressiveoutput with exogenous input, or ARX. Numerous textbooks have beenwritten on this subject, a good discussion of detail is presented in“Applied System Identification” by Jer-Nan Juang (PTR Prentice Hall,1994). As previously noted, details in the methodology of H-infinitycontrol are established in the art and are appreciated from a study of“Essentials of Robust Control” by Kermin Zhou and John C. Doyle(Prentice Hall, 1998).

[0051]FIG. 3 shows step response 302 from step change 304 indicatingposition of control valve 162. This data indicates a first-orderrelationship for the exemplary feedback loop. A discrete time transferfunction is therefore derived as: $\begin{matrix}{\frac{y(z)}{u(z)} = \frac{\beta}{z + \alpha}} & \left( {{Eq}.\quad 1} \right)\end{matrix}$

[0052] Conversion of Equation 1 to sampled time yields:

y(k)=−αy(k−1)+βu(k−1)  (Eq. 2)

[0053] Equation 2 is reformatted into matrix form with α and β inEquation 2 derived from a least-squares solution technique. The furtherderived z-domain ARX model is: $\begin{matrix}{\frac{\hat{P}}{ValveCmd} = \frac{- 0.0371}{z - 0.9775}} & \left( {{Eq}.\quad 3} \right)\end{matrix}$

[0054]FIG. 4 shows a first-order fit 306 to normalized pressure responsedata for the exemplary feedback loop respective to Equation 3. Note thata 100% position for control valve 162 is a completely closedconfiguration for the valve. Since pressure sensor 160 has a non-zeroIC, a bias of approximately-260 kPa is used to transition the ordinateof FIG. 3 to the ordinate of FIG. 4. First-order model prediction 308 asplotted against experimental data 310 is sufficiently accurate with adata minus model variance of σ² _(data-model)=2.8084.

[0055] In order to proceed with continuous-time robust analysis andH-infinity control data schema development, the z-domain transferfunction of Equation 3 is converted to s-domain form as: $\begin{matrix}{\frac{\hat{P}}{ValveCmd} = \frac{- 0.3749}{s + 0.2273}} & \left( {{Eq}.\quad 4} \right)\end{matrix}$

[0056] The conversion to Equation 4 assumes a 1^(st) order transferfunction, a sample rate of 10 Hz, and a zero-order hold.

[0057] The ARX modeling procedure is repeated for several step changesin valve position with Table 1 showing the model coefficients α and βderived from each step change. TABLE 1 System Identification. Start StepStart Time End Step End Time Denominator Numerator Error Model OrderInput Output Valve sec Valve sec alpha beta Variance 1st Valve P 0 13.4100 143.5 −0.982 −0.0319 45.3855 1st Valve P 100 143.5 10 188.1 −0.978−0.0371 2.8084 1st Valve P 10 188.1 0 281.3 −0.985 −0.0260 39.4215 1stValve P 0 281.3 100 284.8 −0.952 −0.0360 0.0319 1st Valve P 100 284.8 20344.9 −0.977 −0.0379 1.0573 1st Valve P 20 344.9 100 441.2 −0.979−0.0363 2.5137 1st Valve P 100 441.2 30 500.6 −0.976 −0.0415 0.7038 1stValve P 30 500.6 100 588.3 −0.978 −0.0340 2.1465 1st Valve P 100 588.340 676.7 −0.973 −0.0412 0.3709 1st Valve P 40 676.7 100 741.3 −0.975−0.0317 1.6924 1st Valve P 100 741.3 50 814.4 −0.972 −0.0365 0.3134 1stValve P 50 814.4 100 871.6 −0.971 −0.0264 0.8339 1st Valve P 100 871.660 923.9 −0.965 −0.0326 0.0914 1st Valve P 60 923.9 100 976.6 −0.957−0.0203 0.3657 1st Valve P 100 976.6 70 1057.0 −0.944 −0.0247 0.0638 1stValve P 70 1057.0 100 1100.6 −0.959 −0.0130 0.1473 1st Valve P 1001100.6 80 1130.8 −0.935 −0.0193 0.0155 1st Valve P 80 1130.8 0 1152.3−0.841 −0.0149 0.0159 1st Valve P 0 1152.3 80 1220.5 −0.982 −0.03827.1789 1st Valve P 80 1220.5 20 1269.3 −0.977 −0.0482 0.6847 1st Valve P20 1269.3 80 1325.5 −0.979 −0.0469 5.0218 1st Valve P 80 1325.5 401373.3 −0.975 −0.0552 0.2094 1st Valve P 40 1373.3 80 1429.4 −0.975−0.0470 2.0239 1st Valve P 80 1429.4 60 1472.4 −0.970 −0.0568 0.0467 1stValve P 60 1472.4 80 1538.0 −0.966 −0.0357 0.2319 1st Valve P 80 1538.0100 1604.1 −0.957 −0.0449 0.0814 FULL DATA Nominal −0.9646 −0.03524.3637 StandardDev 0.0274 0.0111 11.1366

[0058] As should be apparent, the model results are sufficientlyaccurate over numerous open-loop responses of pressure measurements fromsensor 160 to step changes in control valve 162 for deriving a set ofmodels which encompass the essential full range of anticipated behaviorfor fuel cell power system 100 with (a) further confirmation of thefirst-order response nature of the system over the full range ofanticipated behavior and (b) a model minus data variance of less than 10for nearly all cases. Once a set of models is found, a nominal model isderived accounting for modeling uncertainties with uncertaintyparameters and one standard deviation. The nominal discrete model withthe uncertainties (1σ) from Equation 4 and Table 1 is: $\begin{matrix}{\frac{\hat{P}}{ValveCmd} = \frac{- \left( {0.0352 \pm 0.0111} \right)}{z - \left( {0.9646 \pm 0.0274} \right)}} & \left( {{Eq}.\quad 5} \right)\end{matrix}$

[0059] Converting the discrete domain model of Equation 5 to a derivedcontinuous model yields: $\begin{matrix}{\frac{\hat{P}}{ValveCmd} = {\frac{\beta}{s + \alpha} = \frac{- \left( {0.3569 \pm 0.1084} \right)}{s + \left( {0.3645 \pm 0.2882} \right.}}} & \left( {{Eq}.\quad 6} \right)\end{matrix}$

[0060] Once the dynamic response model of Equation 6 is defined, thesystem is formulated into a form consistent with robust analysis. Thisprocess entails (a) “pulling-out” the uncertainties in the control loop,(b) defining weight matrices, and (c) formulating the derived models anddata into a Linear Fractional Transformation (LFT).

[0061]FIG. 5 shows feedback control system model 500 for the pressurecontrol system in fuel cell 122 in a form consistent with robustanalysis. Model block 502, control weight matrix 504 (W_(u)), sensornoise weight matrix 506 (W_(n)), performance weight matrix 508 (W_(e)),system disturbance boundary 512 (W_(do)), system disturbance boundary510 (W_(di)), and Controller 166 (as converted to physical flow viacontrol valve 162) all interrelate as shown to feed fuel (hydrogen) andaffect cathode pressure control within fuel cell 122.

[0062] Turning now to FIG. 6, the uncertainties in the control loop are“pulled out” according to patterned schema 600. With reference toEquation 6, FIG. 6 illustrates how the uncertainties in the dynamicresponse model are removed from the nominal feedback loop. Theconsiderations summarized by FIG. 6 allow the amount of uncertainty inthe feedback loop to be determined by δ_(α) at model block 602 and δ_(β)at model block 604. These parameters are allowed to take a form between±1. A consideration of FIG. 6 indicates that if δ_(α) and δ_(β) are setto zero, a nominal feedback loop results. The offsets are derived fromnormalizing the initial conditions to zero for the ARX model.

[0063] After uncertainties in the control loop have been “pulled out”,weight matrices to provide “weighting” in the real-time computations ofcontroller logic 166 are defined. Three weight matrices are used in theexemplary feedback loop to address three different issues—controlleroutput, noise input, and plant performance output. These matrices areidentified in FIG. 5 as control weight matrix 504 (W_(u)), sensor noiseweight matrix 506 (W_(n)), and performance weight matrix 508 (W_(e)).

[0064] Control weight matrix 504 (W_(u)) is derived from:$\begin{matrix}{W_{u} = \frac{s + 0.1}{{0.1s} + 10}} & \left( {{Eq}.\quad 7} \right)\end{matrix}$

[0065] The control weight matrix, W_(u), is defined to provide fullperformance at low frequencies (<0.1 rad/sec) but reduced performance athigher frequencies (>100 rad/sec). FIG. 7 provides a frequency responseprofile plot 700 of the transfer function in Equation 7.

[0066] Sensor noise weight matrix 506 (W_(n)) is derived from:$\begin{matrix}{W_{n} = \frac{10\left( {s + 10} \right)}{s + 1000}} & \left( {{Eq}.\quad 8} \right)\end{matrix}$

[0067] Sensor noise weight matrix (W_(n)) is defined to provide anessentially low amount of corruption from low frequency (<10 rad/sec)noise with a transition to increased corruption at higher frequencies(>1000 rad/sec). In this regard, pressure sensor 160 is highlysusceptible to high frequency EMI noise. FIG. 8 provides a frequencyresponse profile plot 800 of the transfer function in Equation 8. Notethat, at frequencies greater than 1000 rad/sec, the signal from pressuresensor 160 is highly corrupted by as much as +/−10 kPa.

[0068] Performance noise weight matrix 508 (W_(e)) is derived from:$\begin{matrix}{W_{e} = \frac{s + 0.8}{{0.08s} + 0.0008}} & \left( {{Eq}.\quad 9} \right)\end{matrix}$

[0069] Performance weight matrix 508 (W_(e)) is defined to provide anessentially ideal dynamic response model at near steady-state (<0.1rad/sec) with a transition to a more substantial deviation from theideal response model at higher frequencies. As will be apparent from thebasis of the modeling, the ideal plant is chosen as a simple first ordersystem per model block 502. FIG. 9 provides a frequency response profileplot 900 of the transfer function in Equation 9.

[0070] After “pulling-out” of the uncertainties in the control loop anddefinition of weight matrices has been achieved, the derived models anddata are formulated into a Linear Fractional Transformation (LFT). Theexemplary feedback loop is shown in overall LFT form 1000 in FIG. 10.Difference block 1002, model block 1004, and controller block 1006 allinterrelate as shown in LFT form 1000.

[0071] In the context of FIG. 5 and through use of control weight matrix504 (W_(u)), sensor noise weight matrix 506 (W_(n)), and performanceweight matrix 508 (W_(e)), the following series of equations and therelationship illustrated in FIG. 11 formalize the exemplary closed-loopcontrol system of this specification into LFT form. FIG. 11 shows LFTmatrix 1100 having quadrants A, B, C, and D respectively designatingdomains relative to companion variables with the same A, B, C, and Dprimary symbols in Equations 11, 12, 13, 14, 20, and 21.

e=r−y _(p) −y _(n)  (Eq. 10)

{dot over (x)} _(n) =A _(n) x _(n) +B _(n) n  (Eq. 11)

y _(n) =C _(n) x _(n) +D _(n) n  (Eq. 12)

{dot over (x)} _(n) =A _(n) x _(n) +B _(u) u  (Eq. 13)

y _(u) =C _(u) x _(u) +D _(u) u  (Eq. 14)

{dot over (x)}_(p)=u_(p)  (Eq. 15)

u _(p) =u−100−{overscore (α)}x _(p) −u _(α)  (Eq. 16)

y _(p) ={overscore (β)}x _(p) +u _(β)+100  (Eq. 17)

y _(α) ={circumflex over (α)}x _(p)  (Eq. 18)

y _(β) ={circumflex over (β)}x _(p)  (Eq. 19)

{dot over (x)} _(e) =A _(e) x _(e) +B _(e) e  (Eq. 20)

y _(e) =C _(e) x _(e) +D _(e) e  (Eq. 21)

[y _(α) y _(β) y _(u) y _(e) e] ^(T) =G[u _(α) u _(β)1rnu]  (Eq. 22)

[0072] As should be apparent from the steps leading to theabove-described LFT formalization, concerns related to uncertainty inoperation of fuel cell 122, sensor 160 noise, and responsiveness atdifferent states of operation are effectively incorporated into the LFTformalization.

[0073] Once the system is in LFT form, an H-infinity control data schemais derived. In this regard, the MATLAB μ-analysis toolbox add-onavailable from The Mathworks, Inc. of Natick, Mass. is convenient forsolving the LFT and Algebraic Riccati Equation to calculate a robustcontroller. For the exemplary control loop, the resulting controlstatement for the H-infinity control data schema is $\begin{matrix}{K = \frac{{1.6814s^{3}} - {1854s^{2}} - {169550s} - 87308}{s^{4} + {1007s^{3}} + {12578s^{2}} + {77073s} + 2.1017}} & \left( {{Eq}.\quad 23} \right)\end{matrix}$

[0074] In order to analyze robust stability and nominal performance in asimulator, controller K is incorporated into G to yield

G _(p) =GK  (Eq. 24)

[0075] Wherein: $\begin{matrix}{\left\lbrack \begin{matrix}y_{\alpha} \\y_{\beta} \\y_{u} \\y_{e}\end{matrix}\quad \right\rbrack = {{G_{p}\left\lbrack \quad \begin{matrix}u_{\alpha} \\u_{\beta} \\1 \\r \\n\end{matrix}\quad \right\rbrack} = {\left\lbrack \quad \begin{matrix}G_{p11} & G_{p12} \\G_{p21} & P_{p22}\end{matrix}\quad \right\rbrack\left\lbrack \quad \begin{matrix}u_{\alpha} \\u_{\beta} \\1 \\r \\n\end{matrix}\quad \right\rbrack}}} & \left( {{Eq}.\quad 25} \right)\end{matrix}$

[0076] As will further be appreciated by those of skill, robuststability, nominal performance, and μ-analysis are recommended forapplication to the designed robust controller. Simulated results ofrobust controller use should also be considered for analysis to observethe performance of the control data schema.

[0077] Turning now to FIG. 12 (showing further detail in the H-infinitycontroller), integrator windup compensation block 1202 in controllerlogic 166 (illustrated in FIG. 1) is included to “stall” output fromH-infinity data schema 1006 (including controller K of Equation 11) incases where controller saturation occurs during real-time execution ofschema 1006 by real-time computer 164. In this regard, a controllerderived solely from the H-infinity data schema may have an integralwindup effect. Integrator windup compensation block 1202 effectsreal-time modification of the output from real-time calculations ofH-infinity data schema 1006 to compensate for integrator windup in thecontrol loop and maintain control valve 162 in an immediately responsivestatus. In this regard, controller saturation influence is input viablock 1206 into memory section 1204 in computer 164. Memory section 1204inputs a value to integrator windup compensation block 1202 so thatintegrator windup compensation block 1202 outputs a modifying value forH-infinity data schema 1006.

[0078] A number of benefits are derived from the use of H-infinitycontroller in a fuel cell power system in accordance with the presentinvention. These are appreciated in general comparison to a PID(proportional-integral-derivative) controller. The robust controlapproach of the described H-infinity controller provides superiorperformance in the presence of high frequency feedback noise (e.g.,without limitation, +/−10 kPa high frequency EMI) when compared to astandard PID controller. Since a PID control signal will vary moredramatically than the control signal from an H-infinity controller, asignificantly improved operation of control valve 162 accrues from theuse of the H-infinity controller. A PID control strategy also requiresretuning of its affiliated gains in the event of a change in systemdynamics; this is not needed with the H-infinity controller. Morever,part-to-part actuator variation is also less in the H-infinitycontroller case.

[0079] In the described H-infinity controller, fuel cell uncertaintiesare directly incorporated into the problem formulation. In this regard,a primary problem in control development is that modeled systems thatchange over time frequently render an original dynamic response modelunacceptable for control over the long term which is diminished by useof H-infinity control.

[0080] Signal noise is also incorporated into the formulation of thedescribed H-infinity controller. In this regard, signal corruption istypically quantified according to its frequency response. Filtering ofthese frequencies is applied to the control data schema in theH-infinity controller. This enables deployment of low cost sensors andvalves with a saving to overall system cost. Because the H-infinitycontroller has a high tolerance to EMI, wiring and packaging needs arealso minimized respective to EMI shielding.

[0081] The description of the invention is merely exemplary in natureand, thus, variations that do not depart from the gist of the inventionare intended to be within the scope of the invention. For example, thepreferred embodiment has been described in reference to measuring thepressure of the cathode exhaust stream and controlling the anode feedstream. However, one skilled in the art will appreciate that thelocation of the pressure measurement may be varied. Likewise, the sidesof the fuel cell for measurement (cathode side) and control (anode side)may be interchanged. Such variations are not to be regarded as adeparture from the spirit and scope of the invention.

What is claimed is:
 1. A fuel cell system comprising: a fuel cell withat least one membrane electrode assembly in reactive interface with anoxidant reactant on one face thereof and a fuel reactant on another facethereof; a valve interposed between an oxidant source and said fuel cellto control the flow rate of one of said oxidant and fuel reactants; apressure sensor operable to measure pressure of the other of saidoxidant and fuel reactants; and an H-infinity controller coupled in afeedback loop between said valve and said pressure sensor.
 2. The fuelcell of claim 1 wherein said pressure sensor measures the pressure ofsaid oxidant reactant and said valve controls the flow rate of said fuelreactant.
 3. The fuel cell of claim 1 further comprising an integratorwindup compensator in data communication with said H-infinitycontroller.
 4. The fuel cell of claim 3 further comprising a real-timecomputer wherein said H-infinity controller and said integrator windupcompensator are executed in said real-time computer.
 5. The fuel cell ofclaim 1 wherein said H-infinity controller incorporates a control weightmatrix to provide full performance below a first predetermined frequencyand reduced performance above a second predetermined frequency.
 6. Thefuel cell of claim 5 wherein said control weight matrix is derived fromEquation
 7. 7. The fuel cell of claim 5 wherein said H-infinitycontroller incorporates a control weight matrix according to theresponse profile of FIG.
 7. 8. The fuel cell of claim 1 wherein saidH-infinity controller incorporates a sensor noise weight matrix tocompensate for corruption above a first predetermined frequency.
 9. Thefuel cell of claim 7 wherein said sensor noise weight matrix is derivedfrom Equation
 8. 10. The fuel cell of claim 8 wherein said H-infinitycontroller incorporates a sensor noise weight matrix according to theresponse profile of FIG.
 8. 11. The fuel cell of claim 1 wherein saidH-infinity controller incorporates a performance weight matrix tocompensate for deviation from a steady-state response model above afirst predetermined frequency.
 12. The fuel cell of claim 9 wherein saidperformance weight matrix is derived from Equation
 9. 13. The fuel cellof claim 11 wherein said H-infinity controller incorporates aperformance weight matrix according to the response profile of FIG. 9.14. The fuel cell of claim 1 wherein said H-infinity controllerincorporates a control weight matrix according to the response profileof FIG. 7, a sensor noise weight matrix according to the responseprofile of FIG. 8, and a performance weight matrix according to theresponse profile of FIG.
 9. 15. A method for operating a fuel cellsystem of the type having a fuel cell with at least one membraneelectrode assembly in reactive interface oxidant reactant on one facethereof and a fuel reactant on another face thereof, said methodcomprising: measuring pressure data of one of said oxidant and fuelreactants; deriving a setpoint for a flow rate of the other of saidoxidant and fuel reactor from an H-infinity control model as a functionof said pressure data; and regulating said flow rate based on saidsetpoint.
 16. The method of claim 15 wherein measuring pressure datameasures the pressure of said oxidant reactant, and a setpoint for saidflow rate of said fuel reactant is derived from said oxidant reactantpressure data.
 17. The method of claim 15 further comprising the step ofcompensating for integrator windup in said H-infinity control model. 18.The method of claim 17 wherein deriving a setpoint and compensating forintegrator windup is provided in approximately real time.
 19. The methodof claim 15 wherein deriving a setup includes providing full performancebelow a first predetermined frequency and reduced performance above asecond predetermined frequency.
 20. The method of claim 19 wherein saidcontrol weight matrix is derived from Equation
 7. 21. The method ofclaim 19 wherein said H-infinity controller incorporates a controlweight matrix according to the response profile of FIG.
 7. 22. Themethod of claim 15 wherein deriving a setpoint includes compensating fornoise corruption above a predetermined frequency.
 23. The method ofclaim 22 wherein said sensor noise weight matrix is derived fromEquation
 8. 24. The method of claim 23 wherein said H-infinitycontroller incorporates a sensor noise weight matrix according to theresponse profile of FIG.
 8. 25. The method of claim 15 wherein derivinga setpoint includes compensating for deviation from a steady-state modelabove a predetermined frequency.
 26. The method of claim 25 wherein saidperformance weight matrix is derived from Equation
 9. 27. The method ofclaim 25 wherein said H-infinity controller incorporates a performanceweight matrix according to the response profile of FIG.
 9. 28. Themethod of claim 15 wherein said H-infinity control model incorporates acontrol weight matrix according to the response profile of FIG. 7, asensor noise weight matrix according to the response profile of FIG. 8,and a performance weight matrix according to the response profile ofFIG. 9.